Properties

Label 6027.2.a.y
Level 6027
Weight 2
Character orbit 6027.a
Self dual yes
Analytic conductor 48.126
Analytic rank 0
Dimension 7
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 6027 = 3 \cdot 7^{2} \cdot 41 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6027.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(48.1258372982\)
Analytic rank: \(0\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 861)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 - \beta_{1} ) q^{2} - q^{3} + ( 2 - \beta_{1} + \beta_{2} ) q^{4} -\beta_{5} q^{5} + ( -1 + \beta_{1} ) q^{6} + ( 3 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{8} + q^{9} +O(q^{10})\) \( q + ( 1 - \beta_{1} ) q^{2} - q^{3} + ( 2 - \beta_{1} + \beta_{2} ) q^{4} -\beta_{5} q^{5} + ( -1 + \beta_{1} ) q^{6} + ( 3 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{8} + q^{9} + ( 1 - \beta_{3} - \beta_{5} ) q^{10} + ( 2 + \beta_{4} - \beta_{5} ) q^{11} + ( -2 + \beta_{1} - \beta_{2} ) q^{12} + ( 1 - \beta_{6} ) q^{13} + \beta_{5} q^{15} + ( 3 - 2 \beta_{1} - 2 \beta_{3} + \beta_{4} - \beta_{5} ) q^{16} + ( -2 \beta_{1} - \beta_{3} + \beta_{4} + \beta_{6} ) q^{17} + ( 1 - \beta_{1} ) q^{18} + ( 1 - \beta_{1} + 2 \beta_{2} + \beta_{4} + \beta_{6} ) q^{19} + ( 3 - \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{4} ) q^{20} + ( 1 - \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} - \beta_{6} ) q^{22} + ( \beta_{1} + \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} ) q^{23} + ( -3 + \beta_{1} - \beta_{2} + \beta_{3} ) q^{24} + ( -1 - 2 \beta_{2} + \beta_{3} - \beta_{6} ) q^{25} + ( 3 - \beta_{1} - \beta_{3} + \beta_{4} - 2 \beta_{6} ) q^{26} - q^{27} + ( 1 - \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} ) q^{29} + ( -1 + \beta_{3} + \beta_{5} ) q^{30} + ( -\beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} ) q^{31} + ( 4 + \beta_{2} - \beta_{3} + \beta_{4} - 2 \beta_{5} - \beta_{6} ) q^{32} + ( -2 - \beta_{4} + \beta_{5} ) q^{33} + ( 3 + \beta_{1} + 2 \beta_{2} - \beta_{4} + \beta_{6} ) q^{34} + ( 2 - \beta_{1} + \beta_{2} ) q^{36} + ( 2 + 2 \beta_{2} - 2 \beta_{3} + \beta_{4} ) q^{37} + ( -2 \beta_{1} - \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{6} ) q^{38} + ( -1 + \beta_{6} ) q^{39} + ( 4 - 3 \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} ) q^{40} - q^{41} + ( \beta_{3} + \beta_{4} + \beta_{6} ) q^{43} + ( 5 - \beta_{1} + 3 \beta_{2} - \beta_{3} + \beta_{4} - \beta_{6} ) q^{44} -\beta_{5} q^{45} + ( -3 - \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} + 3 \beta_{6} ) q^{46} + ( -2 + \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} ) q^{47} + ( -3 + 2 \beta_{1} + 2 \beta_{3} - \beta_{4} + \beta_{5} ) q^{48} + ( 3 \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{5} - 2 \beta_{6} ) q^{50} + ( 2 \beta_{1} + \beta_{3} - \beta_{4} - \beta_{6} ) q^{51} + ( 7 - 2 \beta_{1} + \beta_{2} - 3 \beta_{3} + 2 \beta_{4} - 3 \beta_{6} ) q^{52} + ( 3 - 2 \beta_{4} - 2 \beta_{5} + \beta_{6} ) q^{53} + ( -1 + \beta_{1} ) q^{54} + ( 3 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - 2 \beta_{5} - \beta_{6} ) q^{55} + ( -1 + \beta_{1} - 2 \beta_{2} - \beta_{4} - \beta_{6} ) q^{57} + ( -1 + \beta_{1} - \beta_{4} - \beta_{5} + \beta_{6} ) q^{58} + ( 1 - 3 \beta_{1} + \beta_{2} + \beta_{4} - \beta_{5} ) q^{59} + ( -3 + \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{4} ) q^{60} + ( 1 - \beta_{1} + 2 \beta_{2} - 2 \beta_{4} + \beta_{5} + \beta_{6} ) q^{61} + ( 2 + \beta_{3} - \beta_{5} + \beta_{6} ) q^{62} + ( 1 - \beta_{3} - \beta_{4} - 3 \beta_{6} ) q^{64} + ( 4 - \beta_{1} - 2 \beta_{3} + \beta_{4} - 2 \beta_{5} ) q^{65} + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} + \beta_{6} ) q^{66} + ( -1 + \beta_{1} + 2 \beta_{2} + 3 \beta_{3} - \beta_{4} + 2 \beta_{5} - 2 \beta_{6} ) q^{67} + ( -2 \beta_{1} + \beta_{3} - 2 \beta_{4} - \beta_{5} + \beta_{6} ) q^{68} + ( -\beta_{1} - \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} ) q^{69} + ( 1 + \beta_{1} + 2 \beta_{4} + \beta_{5} + \beta_{6} ) q^{71} + ( 3 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{72} + ( 2 - 2 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} ) q^{73} + ( 2 - 3 \beta_{1} + \beta_{2} - 4 \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} ) q^{74} + ( 1 + 2 \beta_{2} - \beta_{3} + \beta_{6} ) q^{75} + ( 6 + \beta_{2} + \beta_{3} - 2 \beta_{5} + 2 \beta_{6} ) q^{76} + ( -3 + \beta_{1} + \beta_{3} - \beta_{4} + 2 \beta_{6} ) q^{78} + ( -2 + 2 \beta_{1} - \beta_{2} + 2 \beta_{3} - 2 \beta_{5} + \beta_{6} ) q^{79} + ( 7 - 3 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} - 3 \beta_{6} ) q^{80} + q^{81} + ( -1 + \beta_{1} ) q^{82} + ( 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} ) q^{83} + ( -2 + 2 \beta_{2} + \beta_{3} - 2 \beta_{4} + 3 \beta_{5} - \beta_{6} ) q^{85} + ( -5 + \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - 3 \beta_{4} + 2 \beta_{5} + \beta_{6} ) q^{86} + ( -1 + \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} ) q^{87} + ( 7 - 5 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} + 3 \beta_{4} - \beta_{6} ) q^{88} + ( -7 + \beta_{1} - \beta_{2} + \beta_{3} - 3 \beta_{4} + 2 \beta_{5} - \beta_{6} ) q^{89} + ( 1 - \beta_{3} - \beta_{5} ) q^{90} + ( -4 + \beta_{1} + \beta_{2} + 2 \beta_{3} - \beta_{4} + \beta_{5} + 5 \beta_{6} ) q^{92} + ( \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} ) q^{93} + ( -7 + 2 \beta_{1} - 3 \beta_{2} + \beta_{3} - 4 \beta_{4} + 2 \beta_{5} + \beta_{6} ) q^{94} + ( 3 \beta_{2} - \beta_{3} + 4 \beta_{5} ) q^{95} + ( -4 - \beta_{2} + \beta_{3} - \beta_{4} + 2 \beta_{5} + \beta_{6} ) q^{96} + ( 6 - \beta_{1} - 4 \beta_{3} + \beta_{4} + 2 \beta_{6} ) q^{97} + ( 2 + \beta_{4} - \beta_{5} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7q + 4q^{2} - 7q^{3} + 8q^{4} - q^{5} - 4q^{6} + 12q^{8} + 7q^{9} + O(q^{10}) \) \( 7q + 4q^{2} - 7q^{3} + 8q^{4} - q^{5} - 4q^{6} + 12q^{8} + 7q^{9} + 3q^{10} + 11q^{11} - 8q^{12} + 7q^{13} + q^{15} + 6q^{16} - 11q^{17} + 4q^{18} - 4q^{19} + 7q^{20} + 6q^{22} + 7q^{23} - 12q^{24} + 2q^{25} + 13q^{26} - 7q^{27} + 4q^{29} - 3q^{30} + 7q^{31} + 18q^{32} - 11q^{33} + 20q^{34} + 8q^{36} - 4q^{38} - 7q^{39} + 9q^{40} - 7q^{41} + q^{43} + 18q^{44} - q^{45} - 17q^{46} - 14q^{47} - 6q^{48} + 19q^{50} + 11q^{51} + 27q^{52} + 23q^{53} - 4q^{54} + 30q^{55} + 4q^{57} - 3q^{58} - 8q^{59} - 7q^{60} + 3q^{61} + 16q^{62} + 6q^{64} + 15q^{65} - 6q^{66} + 3q^{67} - 7q^{69} + 7q^{71} + 12q^{72} + 11q^{73} - 13q^{74} - 2q^{75} + 40q^{76} - 13q^{78} - q^{79} + 43q^{80} + 7q^{81} - 4q^{82} - 10q^{85} - 12q^{86} - 4q^{87} + 10q^{88} - 32q^{89} + 3q^{90} - 19q^{92} - 7q^{93} - 21q^{94} - 8q^{95} - 18q^{96} + 25q^{97} + 11q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{7} - 3 x^{6} - 6 x^{5} + 16 x^{4} + 14 x^{3} - 20 x^{2} - 10 x + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 3 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - 2 \nu^{2} - 3 \nu + 3 \)
\(\beta_{4}\)\(=\)\((\)\( \nu^{6} - 3 \nu^{5} - 4 \nu^{4} + 12 \nu^{3} + 4 \nu^{2} - 8 \nu \)\()/2\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{6} - 3 \nu^{5} - 6 \nu^{4} + 16 \nu^{3} + 12 \nu^{2} - 16 \nu - 4 \)\()/2\)
\(\beta_{6}\)\(=\)\((\)\( -\nu^{6} + 5 \nu^{5} - 2 \nu^{4} - 18 \nu^{3} + 14 \nu^{2} + 14 \nu - 6 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + \beta_{1} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{3} + 2 \beta_{2} + 5 \beta_{1} + 3\)
\(\nu^{4}\)\(=\)\(-\beta_{5} + \beta_{4} + 2 \beta_{3} + 8 \beta_{2} + 10 \beta_{1} + 16\)
\(\nu^{5}\)\(=\)\(\beta_{6} - 3 \beta_{5} + 4 \beta_{4} + 9 \beta_{3} + 21 \beta_{2} + 33 \beta_{1} + 33\)
\(\nu^{6}\)\(=\)\(3 \beta_{6} - 13 \beta_{5} + 18 \beta_{4} + 23 \beta_{3} + 67 \beta_{2} + 83 \beta_{1} + 115\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
2.95889
2.35311
1.19009
0.281557
−0.762978
−1.32604
−1.69463
−1.95889 −1.00000 1.83724 0.513172 1.95889 0 0.318823 1.00000 −1.00525
1.2 −1.35311 −1.00000 −0.169102 −1.31916 1.35311 0 2.93503 1.00000 1.78497
1.3 −0.190092 −1.00000 −1.96387 −2.28332 0.190092 0 0.753499 1.00000 0.434041
1.4 0.718443 −1.00000 −1.48384 3.61951 −0.718443 0 −2.50294 1.00000 2.60041
1.5 1.76298 −1.00000 1.10809 −3.51322 −1.76298 0 −1.57241 1.00000 −6.19372
1.6 2.32604 −1.00000 3.41045 −0.0978008 −2.32604 0 3.28076 1.00000 −0.227488
1.7 2.69463 −1.00000 5.26102 2.08082 −2.69463 0 8.78724 1.00000 5.60704
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 6027.2.a.y 7
7.b odd 2 1 861.2.a.m 7
21.c even 2 1 2583.2.a.u 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
861.2.a.m 7 7.b odd 2 1
2583.2.a.u 7 21.c even 2 1
6027.2.a.y 7 1.a even 1 1 trivial

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(7\) \(-1\)
\(41\) \(1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6027))\):

\( T_{2}^{7} - 4 T_{2}^{6} - 3 T_{2}^{5} + 24 T_{2}^{4} - 7 T_{2}^{3} - 34 T_{2}^{2} + 15 T_{2} + 4 \)
\( T_{5}^{7} + T_{5}^{6} - 18 T_{5}^{5} - 18 T_{5}^{4} + 69 T_{5}^{3} + 57 T_{5}^{2} - 36 T_{5} - 4 \)
\( T_{13}^{7} - 7 T_{13}^{6} - 14 T_{13}^{5} + 114 T_{13}^{4} - 83 T_{13}^{3} - 223 T_{13}^{2} + 294 T_{13} - 72 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - 4 T + 11 T^{2} - 24 T^{3} + 47 T^{4} - 82 T^{5} + 133 T^{6} - 196 T^{7} + 266 T^{8} - 328 T^{9} + 376 T^{10} - 384 T^{11} + 352 T^{12} - 256 T^{13} + 128 T^{14} \)
$3$ \( ( 1 + T )^{7} \)
$5$ \( 1 + T + 17 T^{2} + 12 T^{3} + 144 T^{4} + 72 T^{5} + 874 T^{6} + 366 T^{7} + 4370 T^{8} + 1800 T^{9} + 18000 T^{10} + 7500 T^{11} + 53125 T^{12} + 15625 T^{13} + 78125 T^{14} \)
$7$ \( \)
$11$ \( 1 - 11 T + 93 T^{2} - 550 T^{3} + 2829 T^{4} - 12013 T^{5} + 46873 T^{6} - 160404 T^{7} + 515603 T^{8} - 1453573 T^{9} + 3765399 T^{10} - 8052550 T^{11} + 14977743 T^{12} - 19487171 T^{13} + 19487171 T^{14} \)
$13$ \( 1 - 7 T + 77 T^{2} - 432 T^{3} + 2556 T^{4} - 12040 T^{5} + 50292 T^{6} - 197854 T^{7} + 653796 T^{8} - 2034760 T^{9} + 5615532 T^{10} - 12338352 T^{11} + 28589561 T^{12} - 33787663 T^{13} + 62748517 T^{14} \)
$17$ \( 1 + 11 T + 113 T^{2} + 778 T^{3} + 5015 T^{4} + 26849 T^{5} + 133003 T^{6} + 573436 T^{7} + 2261051 T^{8} + 7759361 T^{9} + 24638695 T^{10} + 64979338 T^{11} + 160443841 T^{12} + 265513259 T^{13} + 410338673 T^{14} \)
$19$ \( 1 + 4 T + 55 T^{2} + 144 T^{3} + 1355 T^{4} + 2800 T^{5} + 30893 T^{6} + 58744 T^{7} + 586967 T^{8} + 1010800 T^{9} + 9293945 T^{10} + 18766224 T^{11} + 136185445 T^{12} + 188183524 T^{13} + 893871739 T^{14} \)
$23$ \( 1 - 7 T + 91 T^{2} - 416 T^{3} + 3674 T^{4} - 12912 T^{5} + 98352 T^{6} - 293570 T^{7} + 2262096 T^{8} - 6830448 T^{9} + 44701558 T^{10} - 116413856 T^{11} + 585707213 T^{12} - 1036251223 T^{13} + 3404825447 T^{14} \)
$29$ \( 1 - 4 T + 150 T^{2} - 604 T^{3} + 10745 T^{4} - 40100 T^{5} + 472193 T^{6} - 1504974 T^{7} + 13693597 T^{8} - 33724100 T^{9} + 262059805 T^{10} - 427197724 T^{11} + 3076672350 T^{12} - 2379293284 T^{13} + 17249876309 T^{14} \)
$31$ \( 1 - 7 T + 177 T^{2} - 1204 T^{3} + 14369 T^{4} - 88973 T^{5} + 693757 T^{6} - 3619024 T^{7} + 21506467 T^{8} - 85503053 T^{9} + 428066879 T^{10} - 1111919284 T^{11} + 5067359727 T^{12} - 6212525767 T^{13} + 27512614111 T^{14} \)
$37$ \( 1 + 118 T^{2} + 22 T^{3} + 7423 T^{4} - 1260 T^{5} + 346623 T^{6} - 120738 T^{7} + 12825051 T^{8} - 1724940 T^{9} + 375997219 T^{10} + 41231542 T^{11} + 8182586926 T^{12} + 94931877133 T^{14} \)
$41$ \( ( 1 + T )^{7} \)
$43$ \( 1 - T + 199 T^{2} + 74 T^{3} + 19163 T^{4} + 15837 T^{5} + 1210385 T^{6} + 921684 T^{7} + 52046555 T^{8} + 29282613 T^{9} + 1523592641 T^{10} + 252991274 T^{11} + 29254680157 T^{12} - 6321363049 T^{13} + 271818611107 T^{14} \)
$47$ \( 1 + 14 T + 230 T^{2} + 2300 T^{3} + 22871 T^{4} + 175728 T^{5} + 1388785 T^{6} + 9267708 T^{7} + 65272895 T^{8} + 388183152 T^{9} + 2374535833 T^{10} + 11223266300 T^{11} + 52749351610 T^{12} + 150909014606 T^{13} + 506623120463 T^{14} \)
$53$ \( 1 - 23 T + 403 T^{2} - 5284 T^{3} + 60362 T^{4} - 576532 T^{5} + 5009514 T^{6} - 38172514 T^{7} + 265504242 T^{8} - 1619478388 T^{9} + 8986513474 T^{10} - 41693301604 T^{11} + 168532783679 T^{12} - 509780305967 T^{13} + 1174711139837 T^{14} \)
$59$ \( 1 + 8 T + 309 T^{2} + 2188 T^{3} + 42973 T^{4} + 268692 T^{5} + 3667649 T^{6} + 19759184 T^{7} + 216391291 T^{8} + 935316852 T^{9} + 8825751767 T^{10} + 26512785868 T^{11} + 220911608391 T^{12} + 337444269128 T^{13} + 2488651484819 T^{14} \)
$61$ \( 1 - 3 T + 253 T^{2} - 1292 T^{3} + 32651 T^{4} - 177077 T^{5} + 2922699 T^{6} - 13311088 T^{7} + 178284639 T^{8} - 658903517 T^{9} + 7411156631 T^{10} - 17888826572 T^{11} + 213682864153 T^{12} - 154561123083 T^{13} + 3142742836021 T^{14} \)
$67$ \( 1 - 3 T + 195 T^{2} - 754 T^{3} + 20178 T^{4} - 83910 T^{5} + 1549642 T^{6} - 6051514 T^{7} + 103826014 T^{8} - 376671990 T^{9} + 6068795814 T^{10} - 15193945234 T^{11} + 263274395865 T^{12} - 271375146507 T^{13} + 6060711605323 T^{14} \)
$71$ \( 1 - 7 T + 323 T^{2} - 2160 T^{3} + 53359 T^{4} - 311321 T^{5} + 5588401 T^{6} - 27474912 T^{7} + 396776471 T^{8} - 1569369161 T^{9} + 19097773049 T^{10} - 54889230960 T^{11} + 582766080373 T^{12} - 896701987447 T^{13} + 9095120158391 T^{14} \)
$73$ \( 1 - 11 T + 433 T^{2} - 3456 T^{3} + 78235 T^{4} - 476001 T^{5} + 8321367 T^{6} - 41221920 T^{7} + 607459791 T^{8} - 2536609329 T^{9} + 30434744995 T^{10} - 98144320896 T^{11} + 897639999769 T^{12} - 1664676489179 T^{13} + 11047398519097 T^{14} \)
$79$ \( 1 + T + 331 T^{2} + 382 T^{3} + 52054 T^{4} + 82270 T^{5} + 5419866 T^{6} + 9127174 T^{7} + 428169414 T^{8} + 513447070 T^{9} + 25664652106 T^{10} + 14878930942 T^{11} + 1018505668069 T^{12} + 243087455521 T^{13} + 19203908986159 T^{14} \)
$83$ \( 1 + 193 T^{2} + 1296 T^{3} + 19473 T^{4} + 190016 T^{5} + 2510665 T^{6} + 13606752 T^{7} + 208385195 T^{8} + 1309020224 T^{9} + 11134408251 T^{10} + 61505984016 T^{11} + 760234844099 T^{12} + 27136050989627 T^{14} \)
$89$ \( 1 + 32 T + 847 T^{2} + 14824 T^{3} + 230117 T^{4} + 2864080 T^{5} + 32862267 T^{6} + 320741968 T^{7} + 2924741763 T^{8} + 22686377680 T^{9} + 162225351373 T^{10} + 930090980584 T^{11} + 4729698353303 T^{12} + 15903401310752 T^{13} + 44231334895529 T^{14} \)
$97$ \( 1 - 25 T + 665 T^{2} - 11462 T^{3} + 182540 T^{4} - 2417768 T^{5} + 28408058 T^{6} - 298204818 T^{7} + 2755581626 T^{8} - 22748779112 T^{9} + 166599329420 T^{10} - 1014722618822 T^{11} + 5710581270905 T^{12} - 20824300123225 T^{13} + 80798284478113 T^{14} \)
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